Its a problem. What I would do is wipe everything down on exterior wood with mineral spirits after moving everything outside (smelly in itself, but tends to go neutral after it flashes off). Apply a coat of paste wax to exterior surfaces afterwards.

I would then coat all the interior wood and bottoms/backs of all pieces with with a spray coat of either Shellac or Lacquer, which effective seals the wood. Most of the smoke odor is 'soaked' into the wood where it is no sealed (the unfinished areas) and by sealing it up, you can cut it down considerably. This will be a lot of work, but the most effective way to deal with it.

I got some smoke remover from a disaster recovery supplier a number of years ago, then put it on the shelf and forgot about it. I looked up the ingredients in the MSDS and it appears to be very similar to mechanics' hand cream, Go-Jo (the original cream not the type with pumice). It's an emulsified mineral spirits. Subsequently, I read a restoration book and the author loves to use DL Blue Label to clean up antiques. It would be worth experimenting.

As far as sealing the unfinished wood, 'thecollector' is right. Shellac is an excellent odor sealer.

If all else fails, find a place that has an ozone booth (moving companies and disaster recovery companies often have these) and pay to have it put in there.

So right now I'm cleaning the hutch base (buffet) (again) with TSP.
It's sitting in the sun to dry.

1 more question: the 5 drawers are lined w/ ?felt or something. It STINKS of smoke! I'm planning on using the spray shellac on the interior/unfinished wood.
--> Can I use this shellac on the felt also?
It's attached to the drawer, so it can't be removed.

Regarding the paste wax:
--> do I need to get one specifically for dark wood or can I use a CLEAR or AMBER paste wax?

I've only found CLEAR, AMBER, or for LIGHT WOOD paste wax. I can order a paste wax for DARK WOOD (see link below).

Thank you for your advice on removing the felt.
I think I have a good chance at saving this diningroom set.
I've removed the felt from 1 drawer and sprayed it w/ shellac. The drawer doesn't smell at all.
I have plans to clean off the front (not sprayed) of the drawer (rest of hutch) w/ TSP again and put it in the sun to "cook off" the rest of the smoke smell.
Then I'll apply the the paste wax.

Generally no. It will dissolve waxes. It can also dissolve Gilsonite, a tar-like substance that is sometimes used as a stain, but usually that's sealed in with a finish.

In the tens of thousands of pieces I've worked on, only one had any reaction to mineral spirits. It was a rustic black piece, probably from Mexico, and mineral spirits pulled off some color. Not enough to effect it, but I stopped there. It was restoration from a fire, so it may have been smoke or some of the colorant.

Varnishes, though thinned with mineral spirits, cure by polymerization and mineral spirits is not a solvent for them, it's a thinner. Big difference. (In the odd world of finish terminology, though, Lacquer Thinner is the solvent for lacquer. Because part of it is alcohol, it will also dissolve shellac.)

I'm going to toss my thoughts into the ring here .
I agree with all of the suggestions , but they are most likely maybe's to your problem . As someone already said the odor has seeped into the wood/finish .
The only 100% foolproof way to rid the furniture of the smell/odor is to refinish it . This removes the old finish as well as a very ,very thin layer of the wood ( during sanding ) . This will effectively remove the odor 100% .
All the other options are more or less going to trap the odor and mask it . Some will be removed from the mineral spirits cleaning , but not likely enough . Especially since you noted you're very sensitive to the smoke odor .

If you've used Pledge on the pieces, you may have already ruined your chances of refinishing. I believe Pledge contains silicone, which is nearly impossible to remove and will cause "fisheyes" in any future finishes. It's best to try to save the current finish and stop using the Pledge.